A081512 a(n) = smallest number which can be expressed as the sum of n of its distinct divisors, or 0 if no such number exists.

1, 0, 6, 12, 24, 24, 48, 60, 84, 120, 120, 120, 180, 180, 240, 360, 360, 360, 360, 672, 720, 720, 720, 840, 840, 1080, 1260, 1260, 1260, 1680, 1680, 1680, 2160, 2520, 2520, 2520, 2520, 2520, 2520, 3360, 4320, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040

Offset: 1

Author

Amarnath Murthy, Mar 27 2003

Keywords

• nonn

Comments

In other words, a(n) is the smallest number m such that m has n distinct divisors d_1, ..., d_n such that d_1+...+d_n = m. (The d_i do not need to be ALL the divisors of m.) For example, a(6) = m = 24, since the divisors of 24 are 1,2,3,4,6,8,12,24, and 1+2+3+4+6+8=24.

a(2) = 0. All other entries are nonzero.

In the following triangle the n-th row gives examples of the n divisors a(1), ..., a(7); a(n) = sum of the n-th row:

1

- -

1 2 3

1 2 3 6

1 2 3 6 12

1 2 3 4 6 8

1 2 3 6 8 12 16

For a given values of a(n) = m, however, there may be more than one way to choose d_1, ..., d_n so that d_1+...+d_n = m.

For n=10, a(10)=120, for example, there are the following equally valid solutions:

[1, 2, 3, 4, 5, 6, 15, 20, 24, 40]

[1, 2, 3, 4, 5, 8, 10, 12, 15, 60]

[1, 2, 3, 4, 5, 8, 12, 15, 30, 40]

[1, 2, 3, 4, 6, 8, 12, 20, 24, 40]

[1, 2, 3, 5, 6, 8, 10, 15, 30, 40]

[1, 2, 3, 5, 8, 10, 12, 15, 24, 40]

[1, 2, 3, 5, 8, 12, 15, 20, 24, 30]

[1, 2, 4, 5, 6, 8, 10, 20, 24, 40]

[1, 2, 4, 6, 8, 10, 15, 20, 24, 30]

[1, 3, 4, 5, 6, 10, 12, 15, 24, 40]

[1, 3, 4, 5, 6, 12, 15, 20, 24, 30]

[1, 3, 4, 5, 8, 10, 15, 20, 24, 30]

[1, 3, 5, 6, 8, 10, 12, 15, 20, 40]

[1, 4, 5, 6, 8, 10, 12, 20, 24, 30]

[2, 3, 4, 5, 6, 8, 10, 12, 30, 40]

[2, 3, 4, 6, 8, 10, 12, 15, 20, 40]

[2, 3, 5, 6, 8, 10, 12, 20, 24, 30]

(These solutions were provided by Jinyuan Wang.)

The lexicographically earliest solution is given as the n-th row of the triangle in A081514. The corresponding value d_n is given in A081513.

The lexicographically earliest solutions are:

..n....m: d_1 d_2 ... d_n

-------------------------

..1....1: 1

..2....0: - -

..3....6: 1, 2, 3

..4...12: 1, 2, 3, 6

..5...24: 1, 2, 3, 6, 12

..6...24: 1, 2, 3, 4, 6, 8

..7...48: 1, 2, 3, 4, 6, 8, 24

..8...60: 1, 2, 3, 4, 5, 10, 15, 20

..9...84: 1, 2, 3, 4, 6, 7, 12, 21, 28

.10..120: 1, 2, 3, 4, 5, 6, 15, 20, 24, 40

...

Examples

			24 is a sum of 6 of its divisors. Namely, 1+2+3+4+6+8=24. Furthermore, 24 is the smallest natural number with at least 6 divisors (not including itself), so it must be the smallest natural number that is a sum of 6 of its divisors.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..552 (first 144 terms from Robert Israel)

Crossrefs

Cf. A033630, A081513, A081514.

See also A081515, A081516, A081517, A081521.

Programs

Maple

A081512 := proc(n) local a, dvs, dset,s,p; if n= 2 then RETURN(0) ; end if; for a from 1 do dvs := numtheory[divisors](a) ; dset := combinat[choose](dvs,n) ; for s in dset do if add(p,p=s) = a then RETURN(a) ; end if; end do; end do: end: for n from 2 do a := A081512(n) ; printf("%d, ",a) ; od: # R. J. Mathar, Nov 11 2008

Mathematica

(* This partly empirical program is just a recomputation of existing data. *)
f[n_, k_] := Module[{c, cc, dd}, dd = Most@ Divisors@k; cc = c[#]& /@ Range@ Length@dd; FindInstance[AllTrue[cc, 0 <= # <= 1&] && cc.dd == k && Total[cc] == n, cc, Integers, 1]];
a[n_] := a[n] = Switch[n, 1, 1, 2, 0, 3, 6, _, For[k = a[n - 1], True, k = k + If[n < 25, 1, 60], If[f[n, k] != {}, Return[k]]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 49}] (* Jean-François Alcover, Oct 21 2024 *)

Extensions

Corrected by Caleb M. Shor (cshor(AT)bates.edu), Sep 26 2007

Extended beyond a(7) by R. J. Mathar, Nov 11 2008

a(16)-a(49) from Max Alekseyev, Jul 27 2009

Edited by N. J. A. Sloane, May 24 2020, following advice from Jinyuan Wang.